We describe the dynamics of the polaritons in the lower polariton branch through the density matrix \(\hat{\rho }\) .
We consider the relaxation and pumping processes in the Born–Markov approximation [1]}.
In this approximation, the density matrix is governed by the master equation in the Lindblad form.
The dissipation of the lower polaritons is described by the Lindblad superoperator [2]}
\({L_{{\rm {diss}}}}\left( {\hat{\rho }} \right) = \sum _{\bf k} {{\gamma _{\bf k}}\left( {{{\hat{a}}_{\bf k}}\hat{\rho }\hat{a}_{\bf k}^\dag - \frac{1}{2}\hat{\rho }\hat{a}_{\bf k}^\dag {{\hat{a}}_{\bf k}} - \frac{1}{2}\hat{a}_{\bf k}^\dag {{\hat{a}}_{\bf k}}\hat{\rho }} \right)} ,\) 
